Simple Gambling
Betting Systems
Conclusions
Betting systems: how not to lose your money
gambling
G. Berkolaiko
Department of Mathematics
Texas A&M University
28 April 2007 / Mini Fair, Math Awareness Month 2007
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambling and Games of Chance
Simple games:
flipping a coin (head / tail)
rolling a die (6 sided)
roulette (18 black, 18 red and 2 green)
Player vs casino
Player predicts outcome and agrees on the wager
If prediction is correct, the player gets money, otherwise he
pays.
Examples:
Coin:
Die:
if head, player wins 1$;
if tail, player loses 1$.
if 5 or above, win 5$;
if 4 or below, lose 3$.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambling and Games of Chance
Simple games:
flipping a coin (head / tail)
rolling a die (6 sided)
roulette (18 black, 18 red and 2 green)
Player vs casino
Player predicts outcome and agrees on the wager
If prediction is correct, the player gets money, otherwise he
pays.
Examples:
Coin:
Die:
if head, player wins 1$;
if tail, player loses 1$.
if 5 or above, win 5$;
if 4 or below, lose 3$.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambling and Games of Chance
Simple games:
flipping a coin (head / tail)
rolling a die (6 sided)
roulette (18 black, 18 red and 2 green)
Player vs casino
Player predicts outcome and agrees on the wager
If prediction is correct, the player gets money, otherwise he
pays.
Examples:
Coin:
Die:
if head, player wins 1$;
if tail, player loses 1$.
if 5 or above, win 5$;
if 4 or below, lose 3$.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambling and Games of Chance
Simple games:
flipping a coin (head / tail)
rolling a die (6 sided)
roulette (18 black, 18 red and 2 green)
Player vs casino
Player predicts outcome and agrees on the wager
If prediction is correct, the player gets money, otherwise he
pays.
Examples:
Coin:
Die:
if head, player wins 1$;
if tail, player loses 1$.
if 5 or above, win 5$;
if 4 or below, lose 3$.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambling and Games of Chance
Simple games:
flipping a coin (head / tail)
rolling a die (6 sided)
roulette (18 black, 18 red and 2 green)
Player vs casino
Player predicts outcome and agrees on the wager
If prediction is correct, the player gets money, otherwise he
pays.
Examples:
Coin:
Die:
if head, player wins 1$;
if tail, player loses 1$.
if 5 or above, win 5$;
if 4 or below, lose 3$.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambling and Games of Chance
Simple games:
flipping a coin (head / tail)
rolling a die (6 sided)
roulette (18 black, 18 red and 2 green)
Player vs casino
Player predicts outcome and agrees on the wager
If prediction is correct, the player gets money, otherwise he
pays.
Examples:
Coin:
Die:
if head, player wins 1$;
if tail, player loses 1$.
if 5 or above, win 5$;
if 4 or below, lose 3$.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambling and Games of Chance
Simple games:
flipping a coin (head / tail)
rolling a die (6 sided)
roulette (18 black, 18 red and 2 green)
Player vs casino
Player predicts outcome and agrees on the wager
If prediction is correct, the player gets money, otherwise he
pays.
Examples:
Coin:
Die:
if head, player wins 1$;
if tail, player loses 1$.
if 5 or above, win 5$;
if 4 or below, lose 3$.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambling and Games of Chance
Simple games:
flipping a coin (head / tail)
rolling a die (6 sided)
roulette (18 black, 18 red and 2 green)
Player vs casino
Player predicts outcome and agrees on the wager
If prediction is correct, the player gets money, otherwise he
pays.
Examples:
Coin:
Die:
if head, player wins 1$;
if tail, player loses 1$.
if 5 or above, win 5$;
if 4 or below, lose 3$.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambling and Games of Chance
Simple games:
flipping a coin (head / tail)
rolling a die (6 sided)
roulette (18 black, 18 red and 2 green)
Player vs casino
Player predicts outcome and agrees on the wager
If prediction is correct, the player gets money, otherwise he
pays.
Examples:
Coin:
Die:
if head, player wins 1$;
if tail, player loses 1$.
if 5 or above, win 5$;
if 4 or below, lose 3$.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambling and Games of Chance
Simple games:
flipping a coin (head / tail)
rolling a die (6 sided)
roulette (18 black, 18 red and 2 green)
Player vs casino
Player predicts outcome and agrees on the wager
If prediction is correct, the player gets money, otherwise he
pays.
Examples:
Coin:
Die:
if head, player wins 1$;
if tail, player loses 1$.
if 5 or above, win 5$;
if 4 or below, lose 3$.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Fair Game
Examples:
Coin tossing with wager 1$ —
Roulette red-black, wager 1$ —
Die: 5 or above win 2$, 4 or below lose 1$ —
Die: 5 or above win 5$, 4 or below lose 3$ —
sum 8 or above win 20$,
—?
2 dice:
sum 7 or below lose 15$
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Fair Game
Examples:
Coin tossing with wager 1$ — fair
Roulette red-black, wager 1$ —
Die: 5 or above win 2$, 4 or below lose 1$ —
Die: 5 or above win 5$, 4 or below lose 3$ —
sum 8 or above win 20$,
—?
2 dice:
sum 7 or below lose 15$
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Fair Game
Examples:
Coin tossing with wager 1$ — fair
Roulette red-black, wager 1$ —
Die: 5 or above win 2$, 4 or below lose 1$ —
Die: 5 or above win 5$, 4 or below lose 3$ —
sum 8 or above win 20$,
—?
2 dice:
sum 7 or below lose 15$
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Fair Game
Examples:
Coin tossing with wager 1$ — fair
Roulette red-black, wager 1$ — unfair to player or subfair
Die: 5 or above win 2$, 4 or below lose 1$ —
Die: 5 or above win 5$, 4 or below lose 3$ —
sum 8 or above win 20$,
—?
2 dice:
sum 7 or below lose 15$
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Fair Game
Examples:
Coin tossing with wager 1$ — fair
Roulette red-black, wager 1$ — unfair to player or subfair
Die: 5 or above win 2$, 4 or below lose 1$ —
Die: 5 or above win 5$, 4 or below lose 3$ —
sum 8 or above win 20$,
—?
2 dice:
sum 7 or below lose 15$
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Fair Game
Examples:
Coin tossing with wager 1$ — fair
Roulette red-black, wager 1$ — unfair to player or subfair
Die: 5 or above win 2$, 4 or below lose 1$ — fair
Die: 5 or above win 5$, 4 or below lose 3$ —
sum 8 or above win 20$,
—?
2 dice:
sum 7 or below lose 15$
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Fair Game
Examples:
Coin tossing with wager 1$ — fair
Roulette red-black, wager 1$ — unfair to player or subfair
Die: 5 or above win 2$, 4 or below lose 1$ — fair
Die: 5 or above win 5$, 4 or below lose 3$ —
sum 8 or above win 20$,
—?
2 dice:
sum 7 or below lose 15$
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Fair Game
Examples:
Coin tossing with wager 1$ — fair
Roulette red-black, wager 1$ — unfair to player or subfair
Die: 5 or above win 2$, 4 or below lose 1$ — fair
Die: 5 or above win 5$, 4 or below lose 3$ — subfair
sum 8 or above win 20$,
—?
2 dice:
sum 7 or below lose 15$
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Fair Game
Examples:
Coin tossing with wager 1$ — fair
Roulette red-black, wager 1$ — unfair to player or subfair
Die: 5 or above win 2$, 4 or below lose 1$ — fair
Die: 5 or above win 5$, 4 or below lose 3$ — subfair
sum 8 or above win 20$,
—?
2 dice:
sum 7 or below lose 15$
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Average: Definition
average = sum of (probability × payment).
(
positive if a win,
payment =
negative if a loss.
probability =
# favourable combinations
# all combinations
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Average: Definition
average = sum of (probability × payment).
(
positive if a win,
payment =
negative if a loss.
probability =
# favourable combinations
# all combinations
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Average: Definition
average = sum of (probability × payment).
(
positive if a win,
payment =
negative if a loss.
probability =
# favourable combinations
# all combinations
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Average: Examples
Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) ×
1
2
=0
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
20
2
average = 1 × 18
38 + (−1) × 38 = − 38 < 0
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
21
15
average = 20 × 15
36 + (−15) × 36 = − 36 < 0
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Average: Examples
Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) ×
1
2
=0
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
20
2
average = 1 × 18
38 + (−1) × 38 = − 38 < 0
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
21
15
average = 20 × 15
36 + (−15) × 36 = − 36 < 0
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Average: Examples
Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) ×
1
2
=0
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
20
2
average = 1 × 18
38 + (−1) × 38 = − 38 < 0
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
21
15
average = 20 × 15
36 + (−15) × 36 = − 36 < 0
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Average: Examples
Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) ×
1
2
=0
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
20
2
average = 1 × 18
38 + (−1) × 38 = − 38 < 0
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
21
15
average = 20 × 15
36 + (−15) × 36 = − 36 < 0
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Average: Examples
Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) ×
1
2
= 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
20
2
average = 1 × 18
38 + (−1) × 38 = − 38 < 0
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
21
15
average = 20 × 15
36 + (−15) × 36 = − 36 < 0
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Average: Examples
Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) ×
1
2
= 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
20
2
average = 1 × 18
38 + (−1) × 38 = − 38 < 0
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
21
15
average = 20 × 15
36 + (−15) × 36 = − 36 < 0
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Average: Examples
Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) ×
1
2
= 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
20
2
average = 1 × 18
38 + (−1) × 38 = − 38 < 0
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
21
15
average = 20 × 15
36 + (−15) × 36 = − 36 < 0
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Average: Examples
Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) ×
1
2
= 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
20
2
average = 1 × 18
38 + (−1) × 38 = − 38 < 0
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
21
15
average = 20 × 15
36 + (−15) × 36 = − 36 < 0
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Average: Examples
Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) ×
1
2
= 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
20
2
average = 1 × 18
38 + (−1) × 38 = − 38 < 0
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
21
15
average = 20 × 15
36 + (−15) × 36 = − 36 < 0
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Average: Examples
Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) ×
1
2
= 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
20
2
average = 1 × 18
38 + (−1) × 38 = − 38 < 0 — subfair!
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
21
15
average = 20 × 15
36 + (−15) × 36 = − 36 < 0
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Average: Examples
Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) ×
1
2
= 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
20
2
average = 1 × 18
38 + (−1) × 38 = − 38 < 0 — subfair!
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
21
15
average = 20 × 15
36 + (−15) × 36 = − 36 < 0
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Average: Examples
Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) ×
1
2
= 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
20
2
average = 1 × 18
38 + (−1) × 38 = − 38 < 0 — subfair!
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
21
15
average = 20 × 15
36 + (−15) × 36 = − 36 < 0
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Average: Examples
Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) ×
1
2
= 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
20
2
average = 1 × 18
38 + (−1) × 38 = − 38 < 0 — subfair!
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
21
15
average = 20 × 15
36 + (−15) × 36 = − 36 < 0
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Average: Examples
Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) ×
1
2
= 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
20
2
average = 1 × 18
38 + (−1) × 38 = − 38 < 0 — subfair!
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
21
15
average = 20 × 15
36 + (−15) × 36 = − 36 < 0
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Average: Examples
Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) ×
1
2
= 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
20
2
average = 1 × 18
38 + (−1) × 38 = − 38 < 0 — subfair!
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
21
15
average = 20 × 15
36 + (−15) × 36 = − 36 < 0
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Average: Examples
Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) ×
1
2
= 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
20
2
average = 1 × 18
38 + (−1) × 38 = − 38 < 0 — subfair!
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
21
15
average = 20 × 15
36 + (−15) × 36 = − 36 < 0 — subfair!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Another type of average
Previous average was the average winning in one round.
Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
18
36
on average you have 9 × 20
38 + 10 × 38 = 9 38 = 10 −
2
38
dollars.
You had 10$ and played roulette. What is you average capital
after round 19?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Another type of average
Previous average was the average winning in one round.
Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
18
36
on average you have 9 × 20
38 + 10 × 38 = 9 38 = 10 −
2
38
dollars.
You had 10$ and played roulette. What is you average capital
after round 19?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Another type of average
Previous average was the average winning in one round.
Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
18
36
on average you have 9 × 20
38 + 10 × 38 = 9 38 = 10 −
2
38
dollars.
You had 10$ and played roulette. What is you average capital
after round 19?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Another type of average
Previous average was the average winning in one round.
Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
18
36
on average you have 9 × 20
38 + 10 × 38 = 9 38 = 10 −
2
38
dollars.
You had 10$ and played roulette. What is you average capital
after round 19?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Another type of average
Previous average was the average winning in one round.
Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
18
36
on average you have 9 × 20
38 + 10 × 38 = 9 38 = 10 −
2
38
dollars.
You had 10$ and played roulette. What is you average capital
after round 19?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Another type of average
Previous average was the average winning in one round.
Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
18
36
on average you have 9 × 20
38 + 10 × 38 = 9 38 = 10 −
2
38
dollars.
You had 10$ and played roulette. What is you average capital
after round 19?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Another type of average
Previous average was the average winning in one round.
Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
18
36
on average you have 9 × 20
38 + 10 × 38 = 9 38 = 10 −
2
38
dollars.
You had 10$ and played roulette. What is you average capital
after round 19?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Another type of average
Previous average was the average winning in one round.
Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
18
36
on average you have 9 × 20
38 + 10 × 38 = 9 38 = 10 −
2
38
dollars.
You had 10$ and played roulette. What is you average capital
after round 19?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Another type of average
Previous average was the average winning in one round.
Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
18
36
on average you have 9 × 20
38 + 10 × 38 = 9 38 = 10 −
2
38
dollars.
You had 10$ and played roulette. What is you average capital
after round 19?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Another type of average
Previous average was the average winning in one round.
Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
18
36
on average you have 9 × 20
38 + 10 × 38 = 9 38 = 10 −
2
38
dollars.
You had 10$ and played roulette. What is you average capital
after round 19?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Another type of average
Previous average was the average winning in one round.
Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
18
36
on average you have 9 × 20
38 + 10 × 38 = 9 38 = 10 −
2
38
dollars.
You had 10$ and played roulette. What is you average capital
after round 19?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Another type of average
Previous average was the average winning in one round.
Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
18
36
on average you have 9 × 20
38 + 10 × 38 = 9 38 = 10 −
2
38
dollars.
You had 10$ and played roulette. What is you average capital
after round 19? 9$.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambler’s Ruin : The Problem
Initially you have n dollars.
You play a game with a wager of 1$ in each round.
In each round, the probability to win is p.
p = 1/2 is fair, p < 1/2 is subfair.
You stop when you get to m dollars or lose all your money.
What is the probability to win the day?
Looking from the other angle, what is the probability of you
financial ruin?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambler’s Ruin : The Problem
Initially you have n dollars.
You play a game with a wager of 1$ in each round.
In each round, the probability to win is p.
p = 1/2 is fair, p < 1/2 is subfair.
You stop when you get to m dollars or lose all your money.
What is the probability to win the day?
Looking from the other angle, what is the probability of you
financial ruin?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambler’s Ruin : The Problem
Initially you have n dollars.
You play a game with a wager of 1$ in each round.
In each round, the probability to win is p.
p = 1/2 is fair, p < 1/2 is subfair.
You stop when you get to m dollars or lose all your money.
What is the probability to win the day?
Looking from the other angle, what is the probability of you
financial ruin?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambler’s Ruin : The Problem
Initially you have n dollars.
You play a game with a wager of 1$ in each round.
In each round, the probability to win is p.
p = 1/2 is fair, p < 1/2 is subfair.
You stop when you get to m dollars or lose all your money.
What is the probability to win the day?
Looking from the other angle, what is the probability of you
financial ruin?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambler’s Ruin : The Problem
Initially you have n dollars.
You play a game with a wager of 1$ in each round.
In each round, the probability to win is p.
p = 1/2 is fair, p < 1/2 is subfair.
You stop when you get to m dollars or lose all your money.
What is the probability to win the day?
Looking from the other angle, what is the probability of you
financial ruin?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambler’s Ruin : The Problem
Initially you have n dollars.
You play a game with a wager of 1$ in each round.
In each round, the probability to win is p.
p = 1/2 is fair, p < 1/2 is subfair.
You stop when you get to m dollars or lose all your money.
What is the probability to win the day?
Looking from the other angle, what is the probability of you
financial ruin?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambler’s Ruin : The Problem
Initially you have n dollars.
You play a game with a wager of 1$ in each round.
In each round, the probability to win is p.
p = 1/2 is fair, p < 1/2 is subfair.
You stop when you get to m dollars or lose all your money.
What is the probability to win the day?
Looking from the other angle, what is the probability of you
financial ruin?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambler’s Ruin : The Formula
Probability theory grew out of this problem.
And it provides the answer:
(
P(n → m) =
where
ρ=
G. Berkolaiko
ρn −1
ρm −1
n
m
if ρ 6= 1,
if ρ = 1,
1−p
.
p
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambler’s Ruin : The Formula
Probability theory grew out of this problem.
And it provides the answer:
(
P(n → m) =
where
ρ=
G. Berkolaiko
ρn −1
ρm −1
n
m
if ρ 6= 1,
if ρ = 1,
1−p
.
p
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambler’s Ruin: Examples
You start with 900$, your goal is 1000$,
Tossing a coin, probability of success is 0.9.
Playing roulette, P = 0.00003.
You start with 100$, and must raise 20,000$ by dawn,
Tossing a coin, probability to get the money is 0.005.
Playing roulette, it is 0.0000...003$
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambler’s Ruin: Examples
You start with 900$, your goal is 1000$,
Tossing a coin, probability of success is 0.9.
Playing roulette, P = 0.00003.
You start with 100$, and must raise 20,000$ by dawn,
Tossing a coin, probability to get the money is 0.005.
Playing roulette, it is 0.0000...003$
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambler’s Ruin: Examples
You start with 900$, your goal is 1000$,
Tossing a coin, probability of success is 0.9.
Playing roulette, P = 0.00003.
You start with 100$, and must raise 20,000$ by dawn,
Tossing a coin, probability to get the money is 0.005.
Playing roulette, it is 0.0000...003$
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambler’s Ruin: Examples
You start with 900$, your goal is 1000$,
Tossing a coin, probability of success is 0.9.
Playing roulette, P = 0.00003.
You start with 100$, and must raise 20,000$ by dawn,
Tossing a coin, probability to get the money is 0.005.
Playing roulette, it is 0.0000...003$
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambler’s Ruin: Examples
You start with 900$, your goal is 1000$,
Tossing a coin, probability of success is 0.9.
Playing roulette, P = 0.00003.
You start with 100$, and must raise 20,000$ by dawn,
Tossing a coin, probability to get the money is 0.005.
Playing roulette, it is 0.0000...003$
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambler’s Ruin: Examples
You start with 900$, your goal is 1000$,
Tossing a coin, probability of success is 0.9.
Playing roulette, P = 0.00003.
You start with 100$, and must raise 20,000$ by dawn,
Tossing a coin, probability to get the money is 0.005.
Playing roulette, it is 0.0000...003$
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Basic Setup and Examples
The Concepts of Fair Game and Average
Gambler’s Ruin
Gambler’s Ruin: Examples
You start with 900$, your goal is 1000$,
Tossing a coin, probability of success is 0.9.
Playing roulette, P = 0.00003.
You start with 100$, and must raise 20,000$ by dawn,
Tossing a coin, probability to get the money is 0.005.
Playing roulette, it is 0.0000...003$ 911 zeros!!!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Betting Systems: Definition
Assume in each round, possible win = possible loss.
Assume you can bet any amount, if you have the money.
Before each round you decide how much you will bet, based
only on the outcomes of the previous rounds.
Simple examples:
Straight Play: bet 1$ every round.
Chicken Play: bet 1$ every round until the first loss, then stop.
In probability theory, the random variable
“capital after round N”
is called martingale (if the game is fair) and supermartingale
(if the game is subfair).
Before being adopted as a mathematical term, martingale was
used to refer to a particular betting system.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Betting Systems: Definition
Assume in each round, possible win = possible loss.
Assume you can bet any amount, if you have the money.
Before each round you decide how much you will bet, based
only on the outcomes of the previous rounds.
Simple examples:
Straight Play: bet 1$ every round.
Chicken Play: bet 1$ every round until the first loss, then stop.
In probability theory, the random variable
“capital after round N”
is called martingale (if the game is fair) and supermartingale
(if the game is subfair).
Before being adopted as a mathematical term, martingale was
used to refer to a particular betting system.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Betting Systems: Definition
Assume in each round, possible win = possible loss.
Assume you can bet any amount, if you have the money.
Before each round you decide how much you will bet, based
only on the outcomes of the previous rounds.
Simple examples:
Straight Play: bet 1$ every round.
Chicken Play: bet 1$ every round until the first loss, then stop.
In probability theory, the random variable
“capital after round N”
is called martingale (if the game is fair) and supermartingale
(if the game is subfair).
Before being adopted as a mathematical term, martingale was
used to refer to a particular betting system.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Betting Systems: Definition
Assume in each round, possible win = possible loss.
Assume you can bet any amount, if you have the money.
Before each round you decide how much you will bet, based
only on the outcomes of the previous rounds.
Simple examples:
Straight Play: bet 1$ every round.
Chicken Play: bet 1$ every round until the first loss, then stop.
In probability theory, the random variable
“capital after round N”
is called martingale (if the game is fair) and supermartingale
(if the game is subfair).
Before being adopted as a mathematical term, martingale was
used to refer to a particular betting system.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Betting Systems: Definition
Assume in each round, possible win = possible loss.
Assume you can bet any amount, if you have the money.
Before each round you decide how much you will bet, based
only on the outcomes of the previous rounds.
Simple examples:
Straight Play: bet 1$ every round.
Chicken Play: bet 1$ every round until the first loss, then stop.
In probability theory, the random variable
“capital after round N”
is called martingale (if the game is fair) and supermartingale
(if the game is subfair).
Before being adopted as a mathematical term, martingale was
used to refer to a particular betting system.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Betting Systems: Definition
Assume in each round, possible win = possible loss.
Assume you can bet any amount, if you have the money.
Before each round you decide how much you will bet, based
only on the outcomes of the previous rounds.
Simple examples:
Straight Play: bet 1$ every round.
Chicken Play: bet 1$ every round until the first loss, then stop.
In probability theory, the random variable
“capital after round N”
is called martingale (if the game is fair) and supermartingale
(if the game is subfair).
Before being adopted as a mathematical term, martingale was
used to refer to a particular betting system.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Betting Systems: Definition
Assume in each round, possible win = possible loss.
Assume you can bet any amount, if you have the money.
Before each round you decide how much you will bet, based
only on the outcomes of the previous rounds.
Simple examples:
Straight Play: bet 1$ every round.
Chicken Play: bet 1$ every round until the first loss, then stop.
In probability theory, the random variable
“capital after round N”
is called martingale (if the game is fair) and supermartingale
(if the game is subfair).
Before being adopted as a mathematical term, martingale was
used to refer to a particular betting system.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Betting Systems: Definition
Assume in each round, possible win = possible loss.
Assume you can bet any amount, if you have the money.
Before each round you decide how much you will bet, based
only on the outcomes of the previous rounds.
Simple examples:
Straight Play: bet 1$ every round.
Chicken Play: bet 1$ every round until the first loss, then stop.
In probability theory, the random variable
“capital after round N”
is called martingale (if the game is fair) and supermartingale
(if the game is subfair).
Before being adopted as a mathematical term, martingale was
used to refer to a particular betting system.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
The Matringale: An Example of a Betting System
Algorithm:
1
2
3
4
5
Bet 1$.
If you lost, bet 2$.
If lost again, bet 4$.
And so forth, doubling the stake each round until you win.
If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
The Matringale: An Example of a Betting System
Algorithm:
1
2
3
4
5
Bet 1$.
If you lost, bet 2$.
If lost again, bet 4$.
And so forth, doubling the stake each round until you win.
If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
The Matringale: An Example of a Betting System
Algorithm:
1
2
3
4
5
Bet 1$.
If you lost, bet 2$.
If lost again, bet 4$.
And so forth, doubling the stake each round until you win.
If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
The Matringale: An Example of a Betting System
Algorithm:
1
2
3
4
5
Bet 1$.
If you lost, bet 2$.
If lost again, bet 4$.
And so forth, doubling the stake each round until you win.
If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
The Matringale: An Example of a Betting System
Algorithm:
1
2
3
4
5
Bet 1$.
If you lost, bet 2$.
If lost again, bet 4$.
And so forth, doubling the stake each round until you win.
If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
The Matringale: An Example of a Betting System
Algorithm:
1
2
3
4
5
Bet 1$.
If you lost, bet 2$.
If lost again, bet 4$.
And so forth, doubling the stake each round until you win.
If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
The Matringale: An Example of a Betting System
Algorithm:
1
2
3
4
5
Bet 1$.
If you lost, bet 2$.
If lost again, bet 4$.
And so forth, doubling the stake each round until you win.
If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
The Matringale: An Example of a Betting System
Algorithm:
1
2
3
4
5
Bet 1$.
If you lost, bet 2$.
If lost again, bet 4$.
And so forth, doubling the stake each round until you win.
If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
The Matringale: An Example of a Betting System
Algorithm:
1
2
3
4
5
Bet 1$.
If you lost, bet 2$.
If lost again, bet 4$.
And so forth, doubling the stake each round until you win.
If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8 = 1.
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
The Matringale: An Example of a Betting System
Algorithm:
1
2
3
4
5
Bet 1$.
If you lost, bet 2$.
If lost again, bet 4$.
And so forth, doubling the stake each round until you win.
If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8 = 1.
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
The Matringale: An Example of a Betting System
Algorithm:
1
2
3
4
5
Bet 1$.
If you lost, bet 2$.
If lost again, bet 4$.
And so forth, doubling the stake each round until you win.
If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8 = 1.
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
The Martingale: A Practitioner’s Testimonial
An excerpt from Casanova’s diary, Venice, 1754:
Playing the martingale, and doubling my stakes
continuously, I won every day during the remainder
of the carnival. [..] I congratulated myself upon
having increased the treasure of my dear mistress [..]
Alas, several days later,
I still played on the martingale, but with such bad
luck that I was soon left without a sequin. As I
shared my property with my mistress, [..] at her
request I sold all her diamonds, losing what I got for
them; she had now only five hundred sequins by her.
There was no more talk of her escaping from the
convent, for we had nothing to live on!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
The Martingale: A Practitioner’s Testimonial
An excerpt from Casanova’s diary, Venice, 1754:
Playing the martingale, and doubling my stakes
continuously, I won every day during the remainder
of the carnival. [..] I congratulated myself upon
having increased the treasure of my dear mistress [..]
Alas, several days later,
I still played on the martingale, but with such bad
luck that I was soon left without a sequin. As I
shared my property with my mistress, [..] at her
request I sold all her diamonds, losing what I got for
them; she had now only five hundred sequins by her.
There was no more talk of her escaping from the
convent, for we had nothing to live on!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
The Martingale: A Practitioner’s Testimonial
An excerpt from Casanova’s diary, Venice, 1754:
Playing the martingale, and doubling my stakes
continuously, I won every day during the remainder
of the carnival. [..] I congratulated myself upon
having increased the treasure of my dear mistress [..]
Alas, several days later,
I still played on the martingale, but with such bad
luck that I was soon left without a sequin. As I
shared my property with my mistress, [..] at her
request I sold all her diamonds, losing what I got for
them; she had now only five hundred sequins by her.
There was no more talk of her escaping from the
convent, for we had nothing to live on!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
The Martingale: A Practitioner’s Testimonial
An excerpt from Casanova’s diary, Venice, 1754:
Playing the martingale, and doubling my stakes
continuously, I won every day during the remainder
of the carnival. [..] I congratulated myself upon
having increased the treasure of my dear mistress [..]
Alas, several days later,
I still played on the martingale, but with such bad
luck that I was soon left without a sequin. As I
shared my property with my mistress, [..] at her
request I sold all her diamonds, losing what I got for
them; she had now only five hundred sequins by her.
There was no more talk of her escaping from the
convent, for we had nothing to live on!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
You Cannot Reverse the Odds
Problem: you need a lot of money if a losing streak is long.
The martingale is a sure thing only with infinite capital!
Theorem
If the game is fair, your average capital remains constant regardless
of the betting system.
Theorem
If the game is subfair, your average capital decreases after every
round regardless of the betting system.
If the game is subfair, you lose no matter how you bet.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
You Cannot Reverse the Odds
Problem: you need a lot of money if a losing streak is long.
The martingale is a sure thing only with infinite capital!
Theorem
If the game is fair, your average capital remains constant regardless
of the betting system.
Theorem
If the game is subfair, your average capital decreases after every
round regardless of the betting system.
If the game is subfair, you lose no matter how you bet.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
You Cannot Reverse the Odds
Problem: you need a lot of money if a losing streak is long.
The martingale is a sure thing only with infinite capital!
Theorem
If the game is fair, your average capital remains constant regardless
of the betting system.
Theorem
If the game is subfair, your average capital decreases after every
round regardless of the betting system.
If the game is subfair, you lose no matter how you bet.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
You Cannot Reverse the Odds
Problem: you need a lot of money if a losing streak is long.
The martingale is a sure thing only with infinite capital!
Theorem
If the game is fair, your average capital remains constant regardless
of the betting system.
Theorem
If the game is subfair, your average capital decreases after every
round regardless of the betting system.
If the game is subfair, you lose no matter how you bet.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
You Cannot Reverse the Odds
Problem: you need a lot of money if a losing streak is long.
The martingale is a sure thing only with infinite capital!
Theorem
If the game is fair, your average capital remains constant regardless
of the betting system.
Theorem
If the game is subfair, your average capital decreases after every
round regardless of the betting system.
If the game is subfair, you lose no matter how you bet.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Example: Gambler’s Ruin with the Martingale
Are betting systems completely worthless then?
Remember example: you have 900$, you want to reach 1000$.
Suppose the game is fair
Using Straight Play, the probability to reach the target is 0.9.
Using the Martingale, the probability remains the same, 0.9.
Suppose you play roulette
Using Straight Play, the probability is 0.00003.
But with the Martingale, it is 0.783!
This is much better than Straight Play!
And not much worse than in a fair game.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Example: Gambler’s Ruin with the Martingale
Are betting systems completely worthless then?
Remember example: you have 900$, you want to reach 1000$.
Suppose the game is fair
Using Straight Play, the probability to reach the target is 0.9.
Using the Martingale, the probability remains the same, 0.9.
Suppose you play roulette
Using Straight Play, the probability is 0.00003.
But with the Martingale, it is 0.783!
This is much better than Straight Play!
And not much worse than in a fair game.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Example: Gambler’s Ruin with the Martingale
Are betting systems completely worthless then?
Remember example: you have 900$, you want to reach 1000$.
Suppose the game is fair
Using Straight Play, the probability to reach the target is 0.9.
Using the Martingale, the probability remains the same, 0.9.
Suppose you play roulette
Using Straight Play, the probability is 0.00003.
But with the Martingale, it is 0.783!
This is much better than Straight Play!
And not much worse than in a fair game.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Example: Gambler’s Ruin with the Martingale
Are betting systems completely worthless then?
Remember example: you have 900$, you want to reach 1000$.
Suppose the game is fair
Using Straight Play, the probability to reach the target is 0.9.
Using the Martingale, the probability remains the same, 0.9.
Suppose you play roulette
Using Straight Play, the probability is 0.00003.
But with the Martingale, it is 0.783!
This is much better than Straight Play!
And not much worse than in a fair game.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Example: Gambler’s Ruin with the Martingale
Are betting systems completely worthless then?
Remember example: you have 900$, you want to reach 1000$.
Suppose the game is fair
Using Straight Play, the probability to reach the target is 0.9.
Using the Martingale, the probability remains the same, 0.9.
Suppose you play roulette
Using Straight Play, the probability is 0.00003.
But with the Martingale, it is 0.783!
This is much better than Straight Play!
And not much worse than in a fair game.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Example: Gambler’s Ruin with the Martingale
Are betting systems completely worthless then?
Remember example: you have 900$, you want to reach 1000$.
Suppose the game is fair
Using Straight Play, the probability to reach the target is 0.9.
Using the Martingale, the probability remains the same, 0.9.
Suppose you play roulette
Using Straight Play, the probability is 0.00003.
But with the Martingale, it is 0.783!
This is much better than Straight Play!
And not much worse than in a fair game.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Example: Gambler’s Ruin with the Martingale
Are betting systems completely worthless then?
Remember example: you have 900$, you want to reach 1000$.
Suppose the game is fair
Using Straight Play, the probability to reach the target is 0.9.
Using the Martingale, the probability remains the same, 0.9.
Suppose you play roulette
Using Straight Play, the probability is 0.00003.
But with the Martingale, it is 0.783!
This is much better than Straight Play!
And not much worse than in a fair game.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Example: Gambler’s Ruin with the Martingale
Are betting systems completely worthless then?
Remember example: you have 900$, you want to reach 1000$.
Suppose the game is fair
Using Straight Play, the probability to reach the target is 0.9.
Using the Martingale, the probability remains the same, 0.9.
Suppose you play roulette
Using Straight Play, the probability is 0.00003.
But with the Martingale, it is 0.783!
This is much better than Straight Play!
And not much worse than in a fair game.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Example: Gambler’s Ruin with the Martingale
Are betting systems completely worthless then?
Remember example: you have 900$, you want to reach 1000$.
Suppose the game is fair
Using Straight Play, the probability to reach the target is 0.9.
Using the Martingale, the probability remains the same, 0.9.
Suppose you play roulette
Using Straight Play, the probability is 0.00003.
But with the Martingale, it is 0.783!
This is much better than Straight Play!
And not much worse than in a fair game.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Example: Gambler’s Ruin with the Martingale
Are betting systems completely worthless then?
Remember example: you have 900$, you want to reach 1000$.
Suppose the game is fair
Using Straight Play, the probability to reach the target is 0.9.
Using the Martingale, the probability remains the same, 0.9.
Suppose you play roulette
Using Straight Play, the probability is 0.00003.
But with the Martingale, it is 0.783!
This is much better than Straight Play!
And not much worse than in a fair game.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Bold Play System
If the game is subfair, every round we lose some (on average).
To minimize losses — minimize rounds!
Algorithm (Bold Play):
1
2
3
Suppose you are within x dollar of your target.
If your can afford a bet of x, do it!
If your cannot afford, bet all you have got!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Bold Play System
If the game is subfair, every round we lose some (on average).
To minimize losses — minimize rounds!
Algorithm (Bold Play):
1
2
3
Suppose you are within x dollar of your target.
If your can afford a bet of x, do it!
If your cannot afford, bet all you have got!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Bold Play System
If the game is subfair, every round we lose some (on average).
To minimize losses — minimize rounds!
Algorithm (Bold Play):
1
2
3
Suppose you are within x dollar of your target.
If your can afford a bet of x, do it!
If your cannot afford, bet all you have got!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Bold Play System
If the game is subfair, every round we lose some (on average).
To minimize losses — minimize rounds!
Algorithm (Bold Play):
1
2
3
Suppose you are within x dollar of your target.
If your can afford a bet of x, do it!
If your cannot afford, bet all you have got!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Bold Play System
If the game is subfair, every round we lose some (on average).
To minimize losses — minimize rounds!
Algorithm (Bold Play):
1
2
3
Suppose you are within x dollar of your target.
If your can afford a bet of x, do it!
If your cannot afford, bet all you have got!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Bold Play System
If the game is subfair, every round we lose some (on average).
To minimize losses — minimize rounds!
Algorithm (Bold Play):
1
2
3
Suppose you are within x dollar of your target.
If your can afford a bet of x, do it!
If your cannot afford, bet all you have got!
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Bold Play: How Does it Fare?
In roulette, going from 900$ to 1000$ will be sucessful with
probability 0.88.
Better than Straight Play (0.00003)
or Martingale (0.78)
and not much worse than in a fair game (0.9).
Raising 20,000$ from 100$ at roulette will succeed with
probability 0.003.
Unlikely
but infinitely better than 0.00...03 with Straight Play
and not much worse than 0.005 chance in a fair game.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Bold Play: How Does it Fare?
In roulette, going from 900$ to 1000$ will be sucessful with
probability 0.88.
Better than Straight Play (0.00003)
or Martingale (0.78)
and not much worse than in a fair game (0.9).
Raising 20,000$ from 100$ at roulette will succeed with
probability 0.003.
Unlikely
but infinitely better than 0.00...03 with Straight Play
and not much worse than 0.005 chance in a fair game.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Bold Play: How Does it Fare?
In roulette, going from 900$ to 1000$ will be sucessful with
probability 0.88.
Better than Straight Play (0.00003)
or Martingale (0.78)
and not much worse than in a fair game (0.9).
Raising 20,000$ from 100$ at roulette will succeed with
probability 0.003.
Unlikely
but infinitely better than 0.00...03 with Straight Play
and not much worse than 0.005 chance in a fair game.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Bold Play: How Does it Fare?
In roulette, going from 900$ to 1000$ will be sucessful with
probability 0.88.
Better than Straight Play (0.00003)
or Martingale (0.78)
and not much worse than in a fair game (0.9).
Raising 20,000$ from 100$ at roulette will succeed with
probability 0.003.
Unlikely
but infinitely better than 0.00...03 with Straight Play
and not much worse than 0.005 chance in a fair game.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Bold Play: How Does it Fare?
In roulette, going from 900$ to 1000$ will be sucessful with
probability 0.88.
Better than Straight Play (0.00003)
or Martingale (0.78)
and not much worse than in a fair game (0.9).
Raising 20,000$ from 100$ at roulette will succeed with
probability 0.003.
Unlikely
but infinitely better than 0.00...03 with Straight Play
and not much worse than 0.005 chance in a fair game.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Bold Play: How Does it Fare?
In roulette, going from 900$ to 1000$ will be sucessful with
probability 0.88.
Better than Straight Play (0.00003)
or Martingale (0.78)
and not much worse than in a fair game (0.9).
Raising 20,000$ from 100$ at roulette will succeed with
probability 0.003.
Unlikely
but infinitely better than 0.00...03 with Straight Play
and not much worse than 0.005 chance in a fair game.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Bold Play: How Does it Fare?
In roulette, going from 900$ to 1000$ will be sucessful with
probability 0.88.
Better than Straight Play (0.00003)
or Martingale (0.78)
and not much worse than in a fair game (0.9).
Raising 20,000$ from 100$ at roulette will succeed with
probability 0.003.
Unlikely
but infinitely better than 0.00...03 with Straight Play
and not much worse than 0.005 chance in a fair game.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Bold Play: How Does it Fare?
In roulette, going from 900$ to 1000$ will be sucessful with
probability 0.88.
Better than Straight Play (0.00003)
or Martingale (0.78)
and not much worse than in a fair game (0.9).
Raising 20,000$ from 100$ at roulette will succeed with
probability 0.003.
Unlikely
but infinitely better than 0.00...03 with Straight Play
and not much worse than 0.005 chance in a fair game.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Betting Systems: Definition and Examples
Can Betting Systems Help?
Bold Play is Optimal
Theorem
For every fixed subfair game, starting capital n and the target m,
Bold Play provides the best chances of success.
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Conclusions
You cannot reverse the odds.
If you must play and have a target,
Bold Play will maximize your chances of reaching it.
How not to lose your money gambling?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Conclusions
You cannot reverse the odds.
If you must play and have a target,
Bold Play will maximize your chances of reaching it.
How not to lose your money gambling?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Conclusions
You cannot reverse the odds.
If you must play and have a target,
Bold Play will maximize your chances of reaching it.
How not to lose your money gambling?
G. Berkolaiko
Betting systems: how not to lose your money gambling
Simple Gambling
Betting Systems
Conclusions
Conclusions
You cannot reverse the odds.
If you must play and have a target,
Bold Play will maximize your chances of reaching it.
How not to lose your money gambling?
DO NOT GAMBLE!
G. Berkolaiko
Betting systems: how not to lose your money gambling